The smoothness of the action of \(G\)  on \(\mathcal {H}^\infty \)  follows from
Theorem REF . This implies that
the natural charts defined by projections of affine
hyperplanes define on \({\mathbb {P}}(\mathcal {H}^\infty )\)  a complex manifold
structure for which \(G\)  acts smoothly by holomorphic maps
(see [1]} for details).
Moreover, the Fubini–Study metric on \({\mathbb {P}}(\mathcal {H})\)  restricts to a
(weak) Kähler metric on \({\mathbb {P}}(\mathcal {H}^\infty )\)  which is invariant
under the \(G\) -action. It is determined by the property that the
projection map \(q\) 
satisfies for any unit vector \(v\)  and \(x,y \in v^\bot \)  the relation
\( \langle T_v(q)x, T_v(q)y\rangle = \langle x,y \rangle . \) 
In particular, we see that \(\Omega \)  defines a weak
symplectic 2-form on \({\mathbb {P}}(\mathcal {H}^\infty )\) 
(cf. [2]}). For \(x \in {\mathfrak {g}}\) , the smooth function
\( H_x([v]) := \frac{1}{i} \frac{\langle {\tt d}\pi (x)v, v \rangle }{\langle v,v\rangle } \) 
on \({\mathbb {P}}(\mathcal {H}^\infty )\) 
now satisfies for \(y \in v^\bot \)  and \(\Vert v\Vert =1\) :
\({\tt d}H_x([v])(T_v(q)y)&= 2\mathop {{\rm Re}}\nolimits \langle -i{\tt d}\pi (x)v,y\rangle = 2 \mathop {{\rm Im}}\nolimits \langle {\tt d}\pi (x)v,y\rangle \\&= -\Omega _{[v]}(T_v(q){\tt d}\pi (x)v, T_v(q)y),\) 

i.e.,
\({\tt d}H_x = i_X \Omega \) 
for the smooth vector field on \({\mathbb {P}}(\mathcal {H}^\infty )\) 
defined by
\(X([v]) := -T_v(q){\tt d}\pi (x)v = \dot{\sigma }(x)([v])\) .
This means that the action of \(G\)  on \({\mathbb {P}}(\mathcal {H}^\infty )\)  is Hamiltonian and
\(\Phi _\pi \colon  {\mathbb {P}}(\mathcal {H}^\infty ) \rightarrow {\mathfrak {g}}^{\prime }\)  is a corresponding momentum
map.
\(\Box \)
The main new point of the preceding theorem is
that it provides for any Banach–Lie group
a natural analytic setup for which the momentum
map \(\Phi _\pi \)  really is a momentum map for a smooth action on
a weak symplectic manifold. The corresponding result
for the action of the Banach–Lie group \(\mathop {\rm U{}}\nolimits (\mathcal {H})\) 
on \({\mathbb {P}}(\mathcal {H})\)  is well-known (cf. [2]}).
A key property of the momentum set \(I_\pi \) 
is that it provides complete information on the extreme spectral
values of the operators \(i{\tt d}\pi (x)\) :
\(\sup ({\rm Spec}(i{\tt d}\pi (x))) = s_{I_\pi }(x)= - \inf \langle I_\pi , x \rangle \quad \mbox{ for } \quad x \in {\mathfrak {g}}\) 
(cf. [4]}).
This relation immediately entails the equivalences
\(x \in I_\pi ^\star \quad \Longleftrightarrow \quad s_{I_\pi }(x) \le 0 \quad \Longleftrightarrow \quad -i{\tt d}\pi (x) \ge 0.\) 
It is now natural to study those representations for which
\(s_\pi \) , resp., the set \(I_\pi \) , contains the most significant
information, which leads to the following concept:
Definition 4.6  A smooth unitary representation \((\pi , \mathcal {H})\)  of a
Lie group \(G\)  is said to be semibounded if its momentum
set \(I_\pi \)  is semi-equicontinuous, i.e., if
\(s_\pi (x) := s_{I_\pi }(x) = \sup ({\rm Spec}(i{\tt d}\pi (x))) \) 
is bounded from above on some non-empty open subset of
\({\mathfrak {g}}\) .

We call \((\pi , \mathcal {H})\)  bounded if
\(I_\pi \)  is equicontinuous.
The representation \((\pi , \mathcal {H})\)  is said to satisfy the
positive energy condition with respect to some \(d \in {\mathfrak {g}}\)  if
\(i{\tt d}\pi (d) \ge 0\) .
It satisfies the positive energy condition with respect to some
convex cone \(C \subseteq {\mathfrak {g}}\) , if
\(i{\tt d}\pi (x) \ge 0\)  holds for each \(x \in C\)  (cf. [5]}).

For a semibounded representation \((\pi , \mathcal {H})\) ,
the domain \(s_\pi ^{-1}({\mathbb {R}})\)  of \(s_\pi \)  is a convex cone
with non-empty interior and \(s_\pi \)  is continuous on this
open cone (cf. [4]}).
Since the momentum set \(I_\pi \)  is invariant under the coadjoint action,
the function \(s_\pi \)  and its domain are
invariant under the adjoint action. This leads to two open
invariant convex cones in \({\mathfrak {g}}\) :
\( W_\pi := \lbrace  x \in {\mathfrak {g}}\colon  s_\pi (x) < \infty \rbrace ^0 = B(I_\pi )^0\quad \mbox{ and } \quad C_\pi := \lbrace  x \in {\mathfrak {g}}\colon  s_\pi (x) \le 0 \rbrace ^0, \) 
where \({}^0\)  denotes the interior of a
set.


Note that (REF ) implies that \(C_\pi \)  is the
interior of the dual cone \(I_\pi ^\star \) .
We collect some basic properties of these two cones:
Proposition 4.7  
For a smooth representation \((\pi , \mathcal {H})\)  of \(G\) ,
the following assertions hold:
(i)
If \(C_\pi \ne \emptyset \) , then \(\pi \)  is semibounded, and if
\(i\mathbf {1} \in {\tt d}\pi ({\mathfrak {g}})\) , then the converse is also true.
(ii)
If \(C_\pi \ne \emptyset \) , then \(H(C_\pi ) = \ker {\tt d}\pi \) .
In particular, \(C_\pi \)  is pointed if
and only if the derived representation \({\tt d}\pi \)  is injective.
(iii)
If \({\mathfrak {h}}\subseteq {\mathfrak {g}}\)  is a subalgebra with the property
that \({\tt d}\pi \vert _{{\mathfrak {h}}}\)  is a bounded representation, then
\({\mathfrak {h}}\subseteq H(W_\pi )\) .
(iv)
Let \(\eta \colon  H \rightarrow G\)  be a morphism of Lie subgroups.
If \(\pi \)  is semibounded, then
\(\pi _H := \pi \circ \eta \)  is semibounded if \(W_\pi \cap \mathop {\bf L{}}\nolimits (\eta ){\mathfrak {h}}\ne \emptyset \) , and then \(I_{\pi _H} = \mathop {\bf L{}}\nolimits (\eta )^*I_\pi \) .
In particular, if \(H \subseteq G\)  is a Lie subgroup,
then the restriction \(\pi _H := \pi \vert _H\)  of \(\pi \) 
to \(H\)  is semibounded
if \(W_\pi \cap {\mathfrak {h}}\ne \emptyset \) , and then \(I_{\pi _H} = I_\pi \vert _{{\mathfrak {h}}}\) .


